Subgroups — C₂×#34036737 #49172149

Order 37504.

1004 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34036737	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22627622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22627619	C₂	1	2	yes	maximal
4	C₂×#22627618	C₂	1	2	yes	maximal
5	C₂×#22627616	C₂	1	2	yes	maximal, complemented
6	C₂×#22627615	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22627614	C₂	1	2	yes	maximal, complemented
8	C₂×#22627604	C₂	1	1	yes	maximal, characteristic
9	C₂×#22627600	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₅₈₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#22627581	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₉₃	C₂²	1	1	yes	characteristic
15	#22627622	C₂²	1	8	yes	complemented
16	#22627619	C₂²	1	16	yes	—
17	#22627618	C₂²	1	16	yes	—
18	#22627616	C₂²	1	16	yes	complemented
19	#22627615	C₂²	1	8	yes	complemented
20	#22627614	C₂²	1	16	yes	complemented
21	#22627604	C₂²	1	8	yes	—
22	#22627600	C₂²	1	8	yes	—
23	C₄×Dic₅₈₆	C₂²	1	8	yes	complemented
24	#22627581	C₂²	1	8	yes	—
25	C₂×C₄⋊Dic₂₉₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
28	C₂×#17989781	C₂²	1	1	yes	characteristic
29	C₂×#17989781	C₂²	1	1	yes	characteristic
30	C₂×#17989781	C₂²	1	2	yes	—
31	C₂×#17989781	C₂²	1	2	yes	—
32	C₂×#17989781	C₂²	1	2	yes	—
33	C₂×#17989781	C₂²	1	2	yes	—
34	C₂×#17989781	C₂²	1	2	yes	—
35	C₂×#17989779	C₂²	1	1	yes	characteristic
36	C₂×#17989779	C₂²	1	1	yes	characteristic
37	C₂×#17989779	C₂²	1	2	yes	—
38	C₂×#17989779	C₂²	1	2	yes	—
39	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₂₉₃	C₂²	1	2	yes	—
41	C₂²×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₂₉₃	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
45	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₂₉₃	C₂²	1	2	yes	—
47	C₂³×D₂₉₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₈₆	—	2	4	no	—
49	C₂×Dic₅₈₆	C₂³	1	4	yes	—
50	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₂₉₃	C₂³	1	2	yes	—
52	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₉₃	C₂³	1	2	yes	—
55	C₂×C₄×D₂₉₃	—	2	4	no	—
56	C₂×C₄×D₂₉₃	C₂³	1	4	yes	—
57	C₄⋊Dic₂₉₃	C₂³	1	4	yes	—
58	C₄⋊Dic₂₉₃	C₂³	1	8	yes	—
59	C₄⋊Dic₂₉₃	C₂³	1	8	yes	—
60	#17989781	C₂³	1	8	yes	—
61	#17989781	C₂³	1	8	yes	—
62	#17989781	C₂³	1	4	yes	—
63	#17989781	C₂³	1	8	yes	—
64	#17989781	C₂³	1	8	yes	—
65	#17989781	C₂³	1	4	yes	—
66	#17989781	C₂³	1	8	yes	—
67	#17989779	C₂³	1	8	yes	—
68	#17989779	C₂³	1	8	yes	—
69	#17989779	C₂³	1	4	yes	—
70	#17989779	C₂³	1	4	yes	—
71	C₄×Dic₂₉₃	C₂³	1	8	yes	—
72	C₄×Dic₂₉₃	C₂³	1	4	yes	—
73	C₂×Q₈×C₂₉₃	—	2	4	no	—
74	C₂×Q₈×C₂₉₃	C₂³	1	4	yes	—
75	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₉₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₂₉₃	C₂³	1	2	yes	—
79	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₂₉₃	C₂³	1	4	yes	—
81	C₄⋊C₄×C₂₉₃	C₂³	1	8	yes	—
82	C₄²×C₂₉₃	C₂³	1	4	yes	—
83	C₄²×C₂₉₃	C₂³	1	8	yes	—
84	C₂²×D₂₉₃	—	2	2	no	—
85	C₂²×D₂₉₃	—	2	2	no	—
86	C₂²×D₂₉₃	—	2	2	no	—
87	C₂²×D₂₉₃	C₂⁴	1	2	yes	—
88	Dic₅₈₆	—	2	8	no	—
89	C₂×Dic₂₉₃	—	2	2	no	—
90	C₂×Dic₂₉₃	—	2	4	no	—
91	C₂×Dic₂₉₃	—	2	2	no	—
92	C₂×Dic₂₉₃	—	2	2	no	—
93	C₂×Dic₂₉₃	—	2	4	no	—
94	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
95	C₂×Dic₂₉₃	C₂⁴	1	4	yes	—
96	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
97	C₂×Dic₂₉₃	C₂⁴	1	4	yes	—
98	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
99	C₄×D₂₉₃	—	2	8	no	—
100	C₂³×C₂₉₃	C₂⁴	1	1	yes	characteristic
101	Q₈×C₂₉₃	—	2	8	no	—
102	C₂×C₄×C₂₉₃	—	2	4	no	—
103	C₂×C₄×C₂₉₃	—	2	2	no	—
104	C₂×C₄×C₂₉₃	—	2	4	no	—
105	C₂×C₄×C₂₉₃	—	2	2	no	—
106	C₂×C₄×C₂₉₃	#41	1	4	yes	—
107	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
109	C₂×C₄×C₂₉₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₂₉₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
112	C₂²×C₂₉₃	C₂×#41	1	2	yes	—
113	C₂²×C₂₉₃	2₋¹⁺⁴	1	2	yes	—
114	C₂²×C₂₉₃	2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₂₉₃	C₂⁵	1	1	yes	characteristic
116	C₂×D₂₉₃	—	2	2	no	—
117	C₂×D₂₉₃	—	2	2	no	—
118	C₂×D₂₉₃	—	2	2	no	—
119	C₂×D₂₉₃	—	4	4	no	—
120	C₄×C₂₉₃	—	2	4	no	—
121	C₄×C₂₉₃	—	2	2	no	—
122	C₄×C₂₉₃	—	2	4	no	—
123	C₄×C₂₉₃	—	2	2	no	—
124	C₄×C₂₉₃	C₂×#41	1	4	yes	—
125	Dic₂₉₃	—	2	2	no	—
126	Dic₂₉₃	—	2	4	no	—
127	Dic₂₉₃	—	2	2	no	—
128	Dic₂₉₃	—	2	4	no	—
129	Dic₂₉₃	—	2	2	no	—
130	C₂×C₂₉₃	#541	1	4	yes	complemented
131	C₂×C₂₉₃	C₂²×#41	1	1	yes	characteristic
132	C₂×C₂₉₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₂₉₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	D₂₉₃	—	4	2	no	—
135	C₂₉₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
136	C₂×#541	—	293	1	no	maximal, Sylow
137	#541	—	293	16	no	—
138	C₂×C₄⋊Q₈	—	293	1	no	—
139	C₂×#126	—	293	2	no	—
140	C₂×#126	—	293	2	no	—
141	C₂×#125	—	293	2	no	—
142	C₂×#125	—	293	1	no	—
143	C₂×#125	—	293	2	no	—
144	C₂×C₂²⋊Q₈	—	293	1	no	—
145	C₂×C₂²⋊Q₈	—	293	1	no	—
146	C₂×C₄×Q₈	—	293	1	no	—
147	C₂×C₄×Q₈	D₂₉₃	1	1	yes	characteristic, complemented
148	C₂×#117	—	293	1	no	—
149	C₂²×Q₈	—	293	1	no	—
150	C₂²×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
151	C₂³×C₄	—	293	1	no	—
152	C₄⋊Q₈	—	293	8	no	—
153	#126	—	293	16	no	—
154	#126	—	293	16	no	—
155	#125	—	293	16	no	—
156	#125	—	293	8	no	—
157	#125	—	293	16	no	—
158	C₂²⋊Q₈	—	293	8	no	—
159	C₂²⋊Q₈	—	293	8	no	—
160	C₄×Q₈	—	293	8	no	—
161	C₄×Q₈	C₂×D₂₉₃	1	8	yes	complemented
162	#117	—	293	8	no	—
163	C₂×C₄⋊C₄	—	293	1	no	—
164	C₂×C₄⋊C₄	—	293	1	no	—
165	C₂×C₄⋊C₄	—	293	1	no	—
166	C₂×C₄⋊C₄	—	293	2	no	—
167	C₂×C₄⋊C₄	—	293	2	no	—
168	C₂×C₄⋊C₄	—	293	2	no	—
169	C₂×C₄⋊C₄	—	293	2	no	—
170	C₂×C₄⋊C₄	—	293	2	no	—
171	C₂×C₄⋊C₄	—	293	2	no	—
172	C₂×C₄⋊C₄	—	293	2	no	—
173	C₂×C₄⋊C₄	C₂×D₂₉₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
175	C₂×C₂²⋊C₄	—	293	1	no	—
176	C₂×C₂²⋊C₄	—	293	1	no	—
177	C₂×C₂²⋊C₄	—	293	2	no	—
178	C₂×C₂²⋊C₄	—	293	2	no	—
179	C₂×C₄²	—	293	1	no	—
180	C₂×C₄²	—	293	2	no	—
181	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×D₂₉₃	1	2	yes	—
183	C₂⁴	—	293	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	293	4	no	—
186	C₂×Q₈	—	586	4	no	—
187	C₂×Q₈	C₂²×D₂₉₃	1	4	yes	—
188	C₂²×C₄	—	293	1	no	—
189	C₂²×C₄	—	293	1	no	—
190	C₂²×C₄	—	293	2	no	—
191	C₂²×C₄	—	293	2	no	—
192	C₂²×C₄	—	293	1	no	—
193	C₂²×C₄	—	293	4	no	—
194	C₂²×C₄	—	586	4	no	—
195	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₂₉₃	1	2	yes	—
198	C₂²×C₄	C₂²×D₂₉₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
200	C₄⋊C₄	—	293	4	no	—
201	C₄⋊C₄	—	293	4	no	—
202	C₄⋊C₄	—	293	4	no	—
203	C₄⋊C₄	—	293	8	no	—
204	C₄⋊C₄	—	293	8	no	—
205	C₄⋊C₄	—	293	8	no	—
206	C₄⋊C₄	—	293	8	no	—
207	C₄⋊C₄	—	293	8	no	—
208	C₄⋊C₄	—	293	8	no	—
209	C₄⋊C₄	—	293	8	no	—
210	C₄⋊C₄	C₂²×D₂₉₃	1	4	yes	—
211	C₄⋊C₄	C₂²×D₂₉₃	1	8	yes	—
212	C₂²⋊C₄	—	293	4	no	—
213	C₂²⋊C₄	—	293	4	no	—
214	C₂²⋊C₄	—	293	8	no	—
215	C₂²⋊C₄	—	293	8	no	—
216	C₄²	—	293	4	no	—
217	C₄²	—	293	8	no	—
218	C₄²	C₂²×D₂₉₃	1	4	yes	—
219	C₄²	C₂²×D₂₉₃	1	8	yes	—
220	C₂³	—	293	2	no	—
221	C₂³	—	586	2	no	—
222	C₂³	—	586	2	no	—
223	C₂³	—	586	2	no	—
224	C₂³	C₂³×D₂₉₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	586	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	293	2	no	—
232	C₂×C₄	—	293	2	no	—
233	C₂×C₄	—	293	4	no	—
234	C₂×C₄	—	293	4	no	—
235	C₂×C₄	—	293	2	no	—
236	C₂×C₄	—	586	2	no	—
237	C₂×C₄	—	586	4	no	—
238	C₂×C₄	—	586	2	no	—
239	C₂×C₄	—	586	2	no	—
240	C₂×C₄	—	586	4	no	—
241	C₂×C₄	—	586	8	no	—
242	C₂×C₄	#17989802	1	4	yes	—
243	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
244	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
245	C₂×C₄	C₂³×D₂₉₃	1	4	yes	—
246	C₂×C₄	C₂³×D₂₉₃	1	4	yes	—
247	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
248	C₂²	—	586	2	no	—
249	C₂²	—	586	2	no	—
250	C₂²	—	586	2	no	—
251	C₂²	—	1172	4	no	—
252	C₂²	C₂×#17989802	1	2	yes	—
253	C₂²	#22627671	1	2	yes	—
254	C₂²	#22627672	1	2	yes	—
255	C₂²	C₂⁴×D₂₉₃	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	586	2	no	—
261	C₄	—	586	4	no	—
262	C₄	—	586	2	no	—
263	C₄	—	586	2	no	—
264	C₄	—	586	4	no	—
265	C₄	C₂×#17989802	1	4	yes	—
266	C₂	—	1172	2	no	—
267	C₂	#34036737	1	4	yes	direct product factor
268	C₂	C₂²×#17989802	1	1	yes	characteristic
269	C₂	C₂×#22627671	1	1	yes	characteristic
270	C₂	C₂×#22627672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₉₃
commutator, cosocle	C₂²×C₂₉₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₉₃
Fitting	C₂×C₄×Q₈×C₂₉₃	C₂
socle	C₂³×C₂₉₃	C₂⁴
prime radical	C₂³×D₂₉₃	C₂³
hypercenter	C₂×C₄×Q₈	D₂₉₃
hypocenter	C₂₉₃	C₂×#541
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #34036737	C₂	direct (N × H)
2. C₂×#22627622	C₂	semidirect
5. C₂×#22627616	C₂	semidirect
6. C₂×#22627615	C₂	semidirect
7. C₂×#22627614	C₂	semidirect
10. C₂×C₄×Dic₅₈₆	C₂	semidirect
12. C₂×C₄×Q₈×C₂₉₃	C₂	semidirect
15. #22627622	C₂²	semidirect
18. #22627616	C₂²	semidirect
19. #22627615	C₂²	semidirect
20. #22627614	C₂²	semidirect
23. C₄×Dic₅₈₆	C₂²	semidirect
42. C₄×Q₈×C₂₉₃	C₂²	semidirect
130. C₂×C₂₉₃	#541	semidirect
135. C₂₉₃	C₂×#541	semidirect
147. C₂×C₄×Q₈	D₂₉₃	semidirect
161. C₄×Q₈	C₂×D₂₉₃	semidirect
267. C₂	#34036737	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#34036737	2
C₂×#22627622	2
C₂×#22627619	2
C₂×#22627618	2
C₂×#22627616	2
C₂×#22627615	2
C₂×#22627614	2
C₂×#22627604	2
C₂×#22627600	2
C₂×C₄×Dic₅₈₆	2
C₂×#22627581	2
C₂×C₄×Q₈×C₂₉₃	2
C₂×#541	293

Subgroups — C2×#34036737 #49172149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#34036737 #49172149